Sparse Linear Models and l1−Regularized 2SLS with High-Dimensional Endogenous Regressors and Instruments

We explore the validity of the 2-stage least squares estimator with l_{1}-regularization in both stages, for linear regression models where the numbers of endogenous regressors in the main equation and instruments in the first-stage equations can exceed the sample size, and the regression coefficients are sufficiently sparse. For this l_{1}-regularized 2-stage least squares estimator, finite-sample performance bounds are established. We then provide a simple practical method (with asymptotic guarantees) for choosing the regularization parameter. We show that this practical method can produce an l_{2}-consistent 2SLS estimator whose rate of convergence can be made as arbitrarily close as the scaling of our finite-sample performance bounds under quite standard conditions.